NIE INSTITUTE OF TECHNOLOGY

DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

Digital Communication
18EC61/17EC61/15EC61

Module-1
Bandpass Signal to Equivalent Lowpass
&
Line Codes
 Module-1

Bandpass Signal to Equivalent Lowpass

&
Line Codes
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❑ Hilbert Transform

❑ Pre-envelopes

❑ Complex envelopes

❑ Canonical representation of bandpass signals

❑ Complex low pass representation of bandpass systems

❑ Complex representation of band pass signals and systems

❑ Line codes: Unipolar, Polar, Bipolar (AMI) and Manchester code and their power spectral densities.

❑ Overview of HDB3, B3ZS, B6ZS

RBT Level : L1, L2, L3

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Hilbert Transform
➢ The Fourier transform is particularly useful for evaluating the frequency content of an energy signal or,
power signal.
➢ It provides the mathematical basis for analyzing and designing frequency-selective filters for the
separation of signals on the basis of their frequency content.
➢ Another method of separating signals is based on phase selectivity, which uses phase shifts between the
pertinent signals to achieve the desired separation.
➢ A phase shift of special interest in this context is that of ±90°.
➢ In particular, when the phase angles of all components of a given signal are shifted by ±90°, the resulting
function of time is known as the Hilbert transform of the signal.
➢ The Hilbert transform is called a quadrature filter; it is so called to emphasize its distinct property of
providing a ±90° phase shift.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Hilbert Transform
➢ To be specific, consider a Fourier transformable signal g(t) with its Fourier transform denoted by G(f).
➢ The Hilbert transform of g(t), which we denote by 𝐠ො 𝐭 , is defined by:

➢ Hilbert transformation is a linear operation.

➢ The inverse Hilbert transform, by means of which the original signal g(t) is linearly recovered from 𝐠ො 𝐭 , is
defined by:

➢ The functions g(t) and 𝐠ො 𝐭 are said to constitute a Hilbert-transform pair.

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Hilbert Transform
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Hilbert Transform
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Hilbert Transform
➢ The definition of the Hilbert transform 𝐠ො 𝐭 , given in (Eq. 1) may be interpreted as the convolution of g(t)
with the time function 1/(πt).
➢ From the convolution theorem listed in Table 2.1 that the convolution of two functions in the time domain is
transformed into the multiplication of their Fourier transforms in the frequency domain.
➢ For the time function 1/(πt), the Fourier-transform pair is:

➢ where sgn(f) is the signum function, defined in the frequency domain as:

4
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Hilbert Transform
෡ 𝒇 of 𝐠ො 𝐭 is given by:
➢ The Fourier transform 𝑮

➢ Equation (5) states that given a Fourier transformable signal g(t), we may obtain the Fourier transform of its
Hilbert transform 𝐠ො 𝐭 by passing g(t) through a linear time-invariant system whose frequency response
is equal to –jsgn(f).
➢ This system may be considered as one that produces a phase shift of –90° for all positive frequencies of
the input signal and +90° degrees for all negative frequencies.
➢ The amplitudes of all frequency components in the signal are unaffected by transmission through the
device.
➢ Such an ideal system is referred to as a Hilbert transformer, or quadrature filter.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Hilbert Transform
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Properties of the Hilbert Transform

➢ The Hilbert transform differs from the Fourier transform in that it operates exclusively in the time
domain.
➢ It has a number of useful properties of its own.
➢ The signal g(t) is assumed to be real valued, which is the usual domain of application of the Hilbert
transform.
➢ For this class of signals, the Hilbert transform has the following properties.
➢ PROPERTY 1: A signal g(t) and its Hilbert transform 𝐠ො 𝐭 have the same magnitude spectrum.

➢ PROPERTY 2 If 𝐠ො 𝐭 is the Hilbert transform of g(t), then the Hilbert transform of 𝐠ො 𝐭 is –g(t).
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Properties of the Hilbert Transform

➢PROPERTY 3 A signal g(t) and its Hilbert transform 𝐠ො 𝐭 are orthogonal over the entire time interval (-∞, ∞).
In mathematical terms, the orthogonality of g(t) and 𝐠ො 𝐭 is described by:

❖ Hilbert Transform of Low-Pass Signal

➢ Consider Figure 2.16a that depicts the Fourier transform of a low-pass signal
g(t), whose frequency content extends from –W to W.
➢ Applying the Hilbert transform to this signal yields a new signal 𝐠ො 𝐭 whose
෡ 𝒇 , is depicted in Figure 2.16b.
Fourier transform, 𝑮
➢ This figure illustrates that the frequency content of a Fourier transformable
signal can be radically changed as a result of Hilbert transformation.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❑ Hilbert Transform

❑ Pre-envelopes

❑ Complex envelopes

❑ Canonical representation of bandpass signals

❑ Complex low pass representation of bandpass systems

❑ Complex representation of band pass signals and systems

❑ Line codes: Unipolar, Polar, Bipolar (AMI) and Manchester code and their power spectral densities.

❑ Overview of HDB3, B3ZS, B6ZS

RBT Level : L1, L2, L3

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Pre-envelopes
➢ The Hilbert transform of a signal is defined for both positive and negative frequencies.
➢ How can we modify the frequency content of a real-valued signal g(t) such that all negative
frequency components are completely eliminated?
➢ The answer to this fundamental question lies in the idea of a complex-valued signal called the pre-envelope
of g(t), formally defined as:
6

➢ Where 𝐠ො 𝐭 is the Hilbert transform of g(t).

➢ According to this definition, the given signal g(t) is the real part of the pre-envelope g+(t), and the Hilbert
transform 𝐠ො 𝒕 is the imaginary part of the pre-envelope.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Pre-envelopes
➢ An important feature of the pre-envelope g+(t) is the behavior of its Fourier transform.
➢ If G+(f) denote the Fourier transform of g+(t).
➢ Then, using Eq (5) and (6) we may write:

➢ Invoking the definition of the signum function given in Eq. (4), we may rewrite Eq. (7) in the equivalent
form as:

where G(0) is the value of G(f) at the origin f = 0

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Pre-envelopes
➢ From the foregoing analysis it is apparent that for a given signal g(t) we may determine its pre-envelope
g+(t) in one of two equivalent procedures.
1. Time-domain procedure: Given the signal g(t), we use (Eq. 6) to compute the pre- envelope g+(t).
2. Frequency-domain procedure: We first determine the Fourier transform G(f) of the signal g(t), then use
(Eq. 8) to determine G+(f), and finally evaluate the inverse Fourier transform of G+(f) to obtain:

➢ Define the pre-envelope for negative frequencies as:

10

➢ The two pre-envelopes g+(t) and g–(t) are simply the complex conjugate of each other, as shown by:

11
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Pre-envelopes
➢ The spectrum of the pre-envelope g+(t) is nonzero only for positive frequencies; hence the use of a plus
sign as the subscript.
➢ the use of a minus sign as the subscript is intended to indicate that the spectrum of the other pre-envelope
g–(t) is nonzero only for negative frequencies, as shown by the Fourier transform:

12

➢ Thus, the pre-envelope g+(t) and g–(t) constitute a complementary pair of complex-valued signals.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Pre-envelopes of Low-Pass Signal

➢ Continuing with the low-pass signal g(t) considered in Hilbert Transform of Low-Pass Signal, Figure 2.17 a
and b depict the corresponding spectra of the pre-envelope g+(t) and the second pre-envelope g–(t),
both of which belong to g(t).
➢ Whereas the spectrum of g(t) is defined for –W ≤ f ≤ W as in Figure 2.16a, we clearly see from Figure 2.17
that the spectral content of g+(t) is confined entirely to 0 ≤ f ≤ W, and
➢ The spectral content of g–(t) is confined entirely to –W ≤ f ≤ 0.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❑ Hilbert Transform

❑ Pre-envelopes

❑ Complex envelopes

❑ Canonical representation of bandpass signals

❑ Complex low pass representation of bandpass systems

❑ Complex representation of band pass signals and systems

❑ Line codes: Unipolar, Polar, Bipolar (AMI) and Manchester code and their power spectral densities.

❑ Overview of HDB3, B3ZS, B6ZS

RBT Level : L1, L2, L3

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Complex Envelopes of Band-Pass Signals

➢ The idea of pre-envelopes basically applies to any real-valued signal, be it of a low-pass or band-pass
kind;
➢ The only requirement is that the signal be Fourier transformable.
➢ We will restrict attention to band-pass signals.
➢ Let the band-pass signal of interest be denoted by s(t) and its Fourier transform be denoted by S(f).
➢ Assuming that the Fourier transform S(f) is essentially confined to a band of frequencies of total extent
2W, centered about some frequency ±fc.

Figure 2.18: (a) Magnitude spectrum of band-pass signal s(t); (b) Magnitude spectrum of pre-envelope s+(t); (c) Magnitude spectrum of complex
envelope ෩
𝑺 𝒕
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Complex Envelopes of Band-Pass Signals

➢ We find that the bandwidth 2W is small compared with fc, so we may refer to the signal s(t) as a
narrowband signal.
➢ Let the pre-envelope of the narrowband signal s(t) be expressed in the form:

13

➢ ෩
𝑺 𝒕 is the complex envelope of the band-pass signal s(t).
➢ Equation (13) may be viewed as the basis of a definition for the complex envelope ෩
𝑺 𝒕 in terms of the pre-
envelope s+(t).
➢ The spectrum of the pre-envelope s+(t) is limited to the positive frequency band fc – W ≤ f ≤ fc + W, as
illustrated in Figure 2.18.c.
➢ Therefore, applying the frequency-shifting property of the Fourier transform to (Eq. 13), we find that the
spectrum of the complex envelope ෩ 𝑺 𝒕 is correspondingly limited to the band –W ≤ f ≤ W and centered at
the origin f = 0, as illustrated in Figure 2.18.b.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Complex Envelopes of Band-Pass Signals

➢ The complex envelope ෩
𝑺 𝒕 of the band-pass signal s(t) is a complex low-pass signal.
➢ The essence of the mapping from the band-pass signal s(t) to the complex low-pass signal ෩
𝑺 𝒕 is
summarized in the following threefold statement:
1. The information content of a modulated signal s(t) is fully preserved in the complex envelope ෩
𝑺 𝒕 .
2. Analysis of the band-pass signal s(t) is complicated by the presence of the carrier frequency fc; in
contrast, the complex envelope ෩
𝑺 𝒕 dispenses with fc, making its analysis simpler to deal with.
3. The use of ෩
𝑺 𝒕 requires having to handle complex notations.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❑ Hilbert Transform

❑ Pre-envelopes

❑ Complex envelopes

❑ Canonical representation of bandpass signals

❑ Complex low pass representation of bandpass systems

❑ Complex representation of band pass signals and systems

❑ Line codes: Unipolar, Polar, Bipolar (AMI) and Manchester code and their power spectral densities.

❑ Overview of HDB3, B3ZS, B6ZS

RBT Level : L1, L2, L3

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Canonical Representation of Band-Pass Signals

➢ By definition, the real part of the pre-envelope s+(t) is equal to the original band-pass signal s(t).
➢ We may therefore express the band-pass signal s(t) in terms of its corresponding complex envelope ෩
𝑺 𝒕
as:
14

➢ where the operator Re[.] denotes the real part of the quantity enclosed inside the square brackets.
➢ Since, in general, ෩
𝑺 𝒕 is a complex-valued quantity, we emphasize this property by expressing it in the
Cartesian form:

15

where sI(t) and sQ(t) are both real-valued low-pass functions

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Canonical Representation of Band-Pass Signals

➢ The original band-pass signal s(t) in the canonical or standard form is represented as:

16

➢ Where sI(t) as the in-phase component of the band-pass signal s(t) and refer to sQ(t) as the quadrature-
phase component or simply the quadrature component of the signal s(t).

Figure: Illustrating an interpretation of the complex envelope ෩

𝑺 𝒕 and its multiplication by exp(j2πfct).
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Canonical Representation of Band-Pass Signals

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Polar Representation of Band-Pass Signals

➢ Equation (15) is the Cartesian form of defining the complex envelope ෩
𝑺 𝒕 of the band-pass signal s(t).
➢ Alternatively, we may define ෩
𝑺 𝒕 in the polar form as:
17
➢ where a(t) and φ(t) are both real-valued low-pass functions.
➢ Based on the polar representation of (Eq.17), the original band-pass signal s(t) is itself defined by:

18

➢ Where a(t) as the natural envelope or simply the envelope of the band-pass signal s(t) and refer to φ(t)
as the phase of the signal.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Relationship Between Cartesian and Polar Representations of Band-Pass Signal

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❑ Hilbert Transform

❑ Pre-envelopes

❑ Complex envelopes

❑ Canonical representation of bandpass signals

❑ Complex low pass representation of bandpass systems

❑ Complex representation of band pass signals and systems

❑ Line codes: Unipolar, Polar, Bipolar (AMI) and Manchester code and their power spectral densities.

❑ Overview of HDB3, B3ZS, B6ZS

RBT Level : L1, L2, L3

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Line codes for the electrical representations of binary data

(a) unipolar
nonreturn-to-zero
(NRZ) signaling

(b) polar NRZ signaling

(c) unipolar return-to-

zero
(RZ) signaling

(d) bipolar RZ signaling

ALTERNATE MARK INVERSION
AMI

(e) split-phase or Manchester code

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Power spectra of line codes

(a) unipolar NRZ signal (b) polar NRZ signal (c) Unipolar RZ signal
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ Power spectra of line codes

(d) bipolar RZ signal (e) Manchester-encoded signal

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

❖ SPECIAL LINE CODES

➢ Long string of zeros can lead to loss of synchronization and timing jitter because there
are no pulses in the waveform for relatively long periods of time.

➢ HDBN & BNZS attempts to improve AMI by substituting a special code of length N for all
strings of N zeros.

➢ HDBN - High-Density Bipolar N

➢ BNZS - Binary N Zero Substitution

 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes
High-Density Bipolar N (HDBN)- HDB3
Consultative Committee for International Telephony and
Telegraphy

➢ This coding algorithm is a CCITT standard recommended by the Conference of European Posts and Telecommunications
Administrations (CEPT), a European standards body.

➢ Whenever there is a string of N + 1 consecutive zeros, they are replaced by a special code of length N + 1 containing AMI violations.

➢ HDB3, is implemented as a CEPT primary digital signal.

➢ String of four consecutive zeros is replaced by either B00V or 000V.

➢ B denotes a pulse obeying the AMI (bipolar) convention and V denotes a pulse violating the AMI convention.
Binary Input
1 0 0 1 1 0 1 1 0 1
sequence

Pulse obeying the AMI

(bipolar) convention

pulse violating the AMI

convention.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes
HDB3 Substitution Rules
Number of B Pulses Since Last Polarity of Last B Pulse Substitution Code Substitution Code Form
Violation

Odd Negative ( − ) 000– 000V

Odd Positive ( + ) 000+ 000V
Even Negative ( − ) +00+ B00V
Even Positive ( + ) –00– B00V

0 0 0 V B 0 0 V B 0 0 V B 0 0 V

0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0

0 + 0 0 0 + - 0 + - 0 0 - + 0 - + 0 0 + - 0 0 - + 0
+V

-V
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

B3ZS Substitution Rules

➢ A string of three consecutive zeros is replaced by either B0V or 00V.

➢ B denotes a pulse obeying the AMI (bipolar) convention and V denotes a pulse violating the AMI
convention.

B3ZS Substitution Rules

Number of B Pulses Since Last Polarity of Last B Pulse Substitution Code Substitution Code Form
Violation

Odd Negative ( − ) 00– 00V

Odd Positive ( + ) 00+ 00V
Even Negative ( − ) +0+ B0V
Even Positive ( + ) –0– B0V
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes
B3ZS Substitution Rules
Number of B Pulses Since Last Polarity of Last B Pulse Substitution Code Substitution Code Form
Violation

Odd Negative ( − ) 00– 00V

Odd Positive ( + ) 00+ 00V
Even Negative ( − ) +0+ B0V
Even Positive ( + ) –0– B0V

0 0 V 0 0 V B 0 V

0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1

0 + 0 0 + 0 - + 0 0 - 0 0 - + - 0 + - + 0 + -
+V

-V
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

B6ZS Substitution Rules

➢ In a B6ZS code, a string of six consecutive zeros is replaced by one of two the special codes according to the
rule:

If the last pulse was positive (+), the special code is: 0+−0−+
If the last pulse was negative (−), the special code is: 0 − + 0 + −

➢ zero indicates a zero voltage level for the bit period; a plus designates a positive pulse; and a minus
indicates a negative pulse.

➢ This special code causes two AMI pulse violations: in its second bit position and in its fifth bit position.
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

B6ZS Substitution Rules

If the last pulse was positive (+), the special code is: 0 +−0−+
If the last pulse was negative (−), the special code is: 0 − + 0 + −

0 V B 0 V B 0 V B 0 V B 0 V B 0 V B
0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1
0 + 0 + - 0 - + 0 0 - 0 - + 0 + - + - 0 0 0 + - 0 - + 0 + - +

+V

-V
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

B8ZS Substitution Rules

If the last pulse was positive (+), the special code is: 000 +−0−+
If the last pulse was negative (−), the special code is: 000 − + 0 + −

0 0 0 V B 0 V B 0 0 0 V B 0 V B
1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0
+ - 0 + 0 0 0 + - 0 - + 0 0 - 0 0 0 - + 0 + - + - 0

+V

-V
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

PROBLEMS
1. A binary data sequence is given below. Assuming first non-zero pulse is positive, sketch the
encoded signal if line coding is i) Bipolar AMI, ii) HDB3. The data sequence is
01100001010000000000101100001011.

2. A binary data sequence is given below. Assuming first non zero pulse is positive, sketch the
encoded signal if line coding is i)B8ZS, ii)B6ZS iii)B3ZS & iv)HDB3.
Data Sequence: 11010000000000100000000110
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

COMPARISION
HDB3 Substitution Rules
Number of B Pulses Since Last Polarity of Last B Pulse Substitution Code Substitution Code Form
Violation

Odd Negative ( − ) 000– 000V

Odd Positive ( + ) 000+ 000V
Even Negative ( − ) +00+ B00V
Even Positive ( + ) –00– B00V

B3ZS Substitution Rules

Number of B Pulses Since Last Polarity of Last B Pulse Substitution Code Substitution Code Form
Violation

Odd Negative ( − ) 00– 00V

Odd Positive ( + ) 00+ 00V
Even Negative ( − ) +0+ B0V
Even Positive ( + ) –0– B0V
 Module 1: Bandpass Signal to Equivalent Lowpass & Line Codes

COMPARISION

B6ZS Substitution Rules

If the last pulse was positive (+), the special code is: 0 +−0−+
If the last pulse was negative (−), the special code is: 0 − + 0 + −

B8ZS Substitution Rules

If the last pulse was positive (+), the special code is: 000 +−0−+
If the last pulse was negative (−), the special code is: 000 − + 0 + −
Thank You
By: Prof. Nitesh K A